Verified Solution • 2025–26 Session moderate

QJEEM21Ma94c7: JEE Main 2021 Physics - Class 11 JEEM21Ma94c7 Solution: If the projection of the vector on the sum of the two vectors…

If the projection of the vector i^+2j^+k^\widehat i + 2\widehat j + \widehat k on the sum of the two vectors 2i^+4j^−5k^2\widehat i + 4\widehat j - 5\widehat k and −λi^+2j^+3k^- \lambda \widehat i + 2\widehat j + 3\widehat k is 1, then λ\lambda is equal to __________.

Step-by-Step Verified Solution

a→=i^+2j^+k^\overrightarrow a = \widehat i + 2\widehat j + \widehat k

b→=(2−λ)i^+6j^−2k^\overrightarrow b = (2 - \lambda )\widehat i + 6\widehat j - 2\widehat k

a→ . b→∣b→∣=1,a→ . b→=12−λ{{\overrightarrow a \,.\,\overrightarrow b } \over {|\overrightarrow b |}} = 1,\overrightarrow a \,.\,\overrightarrow b = 12 - \lambda

(a→ . b→)=∣b→∣2\left( {\overrightarrow a \,.\,\overrightarrow b } \right) = |\overrightarrow b {|^2}

λ\lambda 2 −- 24 λ\lambda + 144 = λ\lambda 2 −- 4 λ\lambda + 4 + 40

20 λ\lambda = 100 ⇒\Rightarrow λ\lambda = 5

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